{
  "id": "1612.04428",
  "version": "v1",
  "published": "2016-12-13T23:26:23.000Z",
  "updated": "2016-12-13T23:26:23.000Z",
  "title": "Limiting spectral distribution for non-Hermitian random matrices with a variance profile",
  "authors": [
    "Nicholas Cook",
    "Walid Hachem",
    "Jamal Najim",
    "David Renfrew"
  ],
  "comment": "89 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "For each $n$, let $A_n=(\\sigma_{ij})$ be an $n\\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\\times n$ random matrix with i.i.d. centered entries of unit variance. We are interested in the asymptotic behavior of the empirical spectral distribution $\\mu_n^Y$ of the rescaled entry-wise product \\[ Y_n = \\left(\\frac1{\\sqrt{n}} \\sigma_{ij}X_{ij}\\right). \\] For our main result, we provide a deterministic sequence of probability measures $\\mu_n$, each described by a family of Master Equations, such that the difference $\\mu^Y_n - \\mu_n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries $\\sigma_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. Examples are also provided where $\\mu_n^Y$ converges to a genuine limit. As an application, we extend the circular law to a class of random matrices whose variance profile is doubly stochastic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-13T23:26:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random matrix",
      "non-hermitian random matrices",
      "limiting spectral distribution",
      "variance profile",
      "standard deviation profiles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 89,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}